Chaos and the Quantum: Limits of Physics Involving the Physics of Limits by William C. McHarris

Dear William C. McHarris,

i thought a long time about possible connections between deterministic chaos and quantum mechanics. I saw some similarities in the behaviour of fractal structures, especially in form of the attractor. That attractor in every fractal pattern led me to the conclusion that for example the interference-pattern in the double-slit experiment with single photons could be a result of a deterministically chaos-process with an attractor as the main picture of the interference-pattern.

I also made my mind up about the Monty-Hall problem.

You can understand the advantage to switch your door after your first choice by extrapolating one run of the game by, for example 999 runs with the same candidate. In this case, if the candidate always changes his initial choice, at the end he wins 666 cars. If we take only *one* run, but extrapolate the game to 999 *doors* with just one car behind them and 998 goats behind the others and the *rules* of the game stay the same, the candidate can be sure to win the car - if he changes his initial choice. The formula for this kind of extrapolation is P = (n - 1)/n where P is the probability to win the car and n is the number of doors. So, for 1000 doors the probability to win the car by changing the initial choice is 999/1000 % .

I thought about, how these two extrapolations could connect to the probability-structure of quantum mechanics but i came to no convincing result. Nonetheless it is my conviction that there could be a deeper connection between the extrapolations of this game and the extrapolations ("probabilities") of quantum mechanics.

But: There is an important detail in the structure of the game that has to be taken into account. The host has to obey two mutually exclusive laws for mutually exclusive cases: In two of three cases - as a result of the probability for the candidate to choose the door with the car with his first choice - the host is *determined* to choose a specail door (namely the door with the second goat behind it). Only in 1/3 of the candidate's initial choices the host can *freely* choose between the two "goat-doors" (this is the case if the candidate's first choice is the door with the car behind it, what could happen in 1/3 of the cases).

Humorously i mentioned the Monty-Hall problem in a response to a comment on my own essay contest contribution with the title "To be or not to be strictly deterministic?". Feel free to check it out if you are interested in the connection between the two seemingly mutual exclusive ways of explanations of nature, "deterministically" versus "free-will-governed". I assume that both alternatives can be merged in a similar way the host of the game has to handle them.

Best,

Stefan Weckbach

Dear William,

As a former nonlinear dynamics practitioner, I had enjoyed your essay very much. Did you consider 't Hooft's approach of deriving QM as an emergent theory from a chaotic deterministic theory?

Regards,

Florin

Have you seen Tim Plamer's work on invariant sets, fractal attractors and QM?

http://arxiv.org/ftp/arxiv/papers/0812/0812.1148.pdf

A nice laymen's version of the arguments is given in the excellent book by Ian Stewart entitled "Does God Play Dice?"

Yours in science,

RL Oldershaw

www.amherst.edu/~rloldershaw

http://arxiv.org/a/oldershaw_r_1

Oops, that's Tim Palmer.

Dear William C. McHarris,

there is a brand new paper about experimental results concerning chaos and quantum mechanics. It is titled "Butterfly effect gets entangled: Chaotic behaviour emerges from quantum entanglement" and can be found under the breaking news as a twitter-link in the fqxi-community.

Perhaps it could be of interest for your topic.

Best,

Stefan Weckbach

Mr. McHarris,

Thank you for an interesting, well written, thought-provoking essay.

Fwiw, there's another nice treatment of the so-called Monte Hall Paradox in 'The Drunkard's Walk: How Randomness Rules Our Lives' by Leonard Mlodinow.

Cheers

Dr. McHarris,

Very interesting.

Are you familiar with the work of Arjendu Pattanayak? Good stuff at Arxiv.

I love this understated (you could almost say tongue-in-cheek) take by the Templeton Prize's own Bernard d'Espagnat (quote):

Hence, to sum up, contrary to a view that seems to be widespread it appears that whoever considers himself to be an adept of objectivist realism cannot logically claim that the phenomena related to chaos violate determinism just because they are "chaotic." Let it however immediately be added that, anyhow, such considerations are, to a large extent, academic since they bear on classical physics and its laws. It is well known that, at the level of anything that might deserve the name "ultimate reality" or even just that of "microscopic reality," these laws are violated (we know that, at the microscopic level, only the quantum ones are correct). Elementary as it is, this observation severely diminishes the pertinence of the, sometimes uttered, statement according to which the advent of the theory of chaos constitutes one of the most important conceptual upheavals that ever took place in physics. But of course this is not to say that the theory in question is uninteresting as far as basic ideas are concerned. Quite on the contrary it has the considerable interest of showing that within classical physics there are phenomena that imitate intrinsically random ones to such an extent that they are operationally indistinguishable from the latter. ("On Physics and Philosophy" page 319)

Greetings NM,

I would say the Bernard d'Espagnat cannot think outside the quantum box.

His theoretical assumptions, derived from a thoroughly heuristic quantum mechanics and dubiously interpreted experiments, may be acceptable to the current crop of theoretical physicists, but I do not think they will hold up in the near future.

Nonlinear dynamical systems theory will eventually explain all the "weirdness" of quantum mechanics in a manner that is fully causal and deterministic. The revolution is already well underway.

Welcome to the new paradigm,

RLO

www.amherst.edu/~rloldershaw

RLO,

I think Bd'E is more than a knee-jerk sycophant and I'm sure Dr. McHenry with his declared respect for both Bell and Aspect would agree. Also unless I'm mistaken Espagnat's the guy who first realized Bell's inequality can be experimentally tested in the macroworld and without violation. That requires a certain offbeat imagination, indeed evinces a refreshing playfulness. Even demonstrates a mild readiness to flirt with danger.

I'm cautious about trumpeting paradigms, new, old or re-treaded. For a demonstration see the Calude and Svozil thread if you haven't already. They may be getting ready to cream me.

DAMN. That's McHARRIS.

Maximum apology.

NM,

Let me clarify.

In the quotation of d'E-nat which you posted, the sentence beginning; "It is well known..." is fatuous and unscientific in its arrogant assumptions that the Aspect-type experiments of Bell's inequality theorems are fully understood and definitively explained.

There are serious scientists, like Tim Palmer and J. Christian, among many others, including Dr. Mc Harris, who have published radically different assessments.

For d'E-nat to summarily dismiss nonlinear dynamical systems theory as a fundamental approach to understanding atoms and quantum "weirdness" is quite regrettable. I predict that he will be forced to revise his opinions considerably in the next 10 years, or less.

Yours in science,

RLO

www.amherst.edu/~rloldershaw

Dear William,

thank you for sharing your ideas. It's an intriguing observation that the emergence of a discrete number of modes in nonlinear dynamics resemble the discretenss of observables in quantum theory. (Although note that the reason why the Tacoma Narrows bridge collapsed was in fact not resonance, but a self-exiciting oscillation.) Also the universality that you mention is present in very much the same way in quantum field theory and statistical field theory, where the process of renormalization leads to a phenomenology at phase transitions which only depends on the field content and the symmetries of the theory, but not on its particular dynamics.

However, it is well-known that quantum theory cannot be an emergent large-scale effective description of an underlying classical system. This is prevented, for example, by the Kochen-Specker theorem. But let me just mention here a few remarks about your essay which should underline this.

(1) You mention that exponential decay can be modelled by an underlying chaotic system with exit states. Under the ergodic hypothesis, this is not surprising since both decays are governed by the differential equation

[math]\dot{N}(t)=-\lambda N(t)\,.[/math]

But now, quantum-mechanical systems can show other types of decay behavior. For example, an exponential decay modulated by an oscillation -- see for example figure 3(a) in this paper.

(2) Concerning Bell inequalities, you speculate that quantum non-locality may possibly be explained in terms of classical conditional. This is incorrect. Classical random correlations also violate the Bell inequalities. A

http://en.wikipedia.org/wiki/Local_hidden_variable_theory#Local_hidden_variables_and_the_Bell_tests">local hidden variable theory](https://

http://en.wikipedia.org/wiki/Local_hidden_variable_theory#Local_hidden_variables_and_the_Bell_tests) is defined in such a way that it does allow classical random correlations. They also satisfy all Bell inequalities due to linearity the latter: suppose you have a model given by some chaotic system which predicts that there is a probability of 3/4 for one set of outcomes, and a probability of 1/4 for another set of outcomes. For each set of outcomes, the CHSH quantity will be +2 or -2; let's say it is +2 in the first case and -2 in the second case. Then due to linearity of expectation values, the overall CHSH inequality for the total ensemble is

[math]\frac{3}{4}\cdot (+2) + \frac{1}{4}\cdot (-2) = +1[/math]

Hence, the CHSH inequality still holds. Of course the same reasoning works with any other numbers defining a statistical ensemble of deterministic hidden variable theories.

(3) Quantum correlations and quantum interference are of a very special kind. For example, quantum probabilities are quadratic in the amplitudes, so that the interference terms are linear in each component. (Try Sorkin's work on quantum measures for detailed accounts of this.) Hence for explaining quantum theory, it is not sufficient to explain why not all correlations are classical; you also need to explain why no correlations are stronger than quantum!

The think which I like on the approach of applying non-linear dynamics as hidden variable theory for quantum mechanics is the fact that then it would be rather general theory. For example, most systems biologists already accepted that limit cycles lie behind the stability of intracellular metabolism. Equally well population biologists work similarly, the metheorologists as well, many astronomers, economists .... In case that molecules are stable for the same reasons we may be approaching new universal theory.